EMISSIONS, CONCENTRATIONS, & TEMPERATURE:

A TIME SERIES ANALYSIS

ROBERT K. KAUFMANN1, HEIKKI KAUPPI2 and JAMES H. STOCK3

1Center for Energy & Environmental Studies, Boston University, Boston, MA 02215E-mail: Kaufmann@bu.edu

2Department of Economics, University of Helsinki, P.O. Box 17 (Arkadiankatu 7),FIN-00014, Finland

3Kennedy School of Government, Harvard University, Cambridge, MA 02138

Abstract. We use recent advances in time series econometrics to estimate the relation among emis-sions of CO2 and CH4, the concentration of these gases, and global surface temperature. These modelsare estimated and specified to answer two questions; (1) does human activity affect global surfacetemperature and; (2) does global surface temperature affect the atmospheric concentration of carbondioxide and/or methane. Regression results provide direct evidence for a statistically meaningful rela-tion between radiative forcing and global surface temperature. A simple model based on these resultsindicates that greenhouse gases and anthropogenic sulfur emissions are largely responsible for thechange in temperature over the last 130 years. The regression results also indicate that increases insurface temperature since 1870 have changed the flow of carbon dioxide to and from the atmospherein a way that increases its atmospheric concentration. Finally, the regression results for methane hintthat higher temperatures may increase its atmospheric concentration, but this effect is not estimatedprecisely.

1. Introduction

Evidence for the effect of human activity on climate comes from two sources:experiments run by climate models and statistical analyses of historical data. Theability of climate models to simulate the spatial/temporal temperature record isimproved by including the radiative forcing of greenhouse gases and troposphericsulfates (Mitchell and Karoly, 2001; Wigley et al., 1998; Santer et al., 1996; Mitchellet al., 1995). Consistent with these results, statistical analyses indicate that there isa link between surface temperature and the radiative forcing of greenhouse gasesand anthropogenic sulfur emissions (Kaufmann and Stern, 1997, 2002; Stern andKaufmann, 2000; Tol and de Vos, 1998).

The interpretation of statistical results is complicated by stochastic trends in thehistorical time series, which can generate spurious regression results, and by thepossibility that surface temperature and the atmospheric concentrations of CO2 andCH4 are determined jointly, which can generate simultaneous equation bias. Here,we address these potential sources of error by estimating the relationship amonganthropogenic emissions of CO2 and CH4, the concentration of these gases, andglobal surface temperature. To avoid spurious regression results, we use the dynamicordinary least squares (DOLS) estimator developed by Stock and Watson (1993).

Climatic Change (2006) 77: 249–278DOI: 10.1007/s10584-006-9062-1 c© Springer 2006

250 R. K. KAUFMANN ET AL.

To avoid simultaneous equation bias, we use an instrumental variables procedure.These techniques are used to estimate equations that are specified to answer twoquestions; (1) does human activity affect global surface temperature and; (2) doesglobal surface temperature affect the atmospheric concentration of carbon dioxideand/or methane. The results provide direct evidence that since 1870: (1) humanactivity is largely responsible for the increase in global surface temperature and;(2) higher surface temperature has increased the atmospheric concentration of CO2

and perhaps CH4.

2. Methodology

We build a simplified model of the climate system that includes statistically esti-mated equations for three endogenous variables: global surface temperature andthe atmospheric concentrations of CO2 and CH4. Each of these equations spec-ify explanatory variables that include another endogenous variable and variablesexogenous to the system (Box 1).

The specifications and techniques that are used to estimate the temperature andconcentration equations are determined by the presence of stochastic trends in thedata and the possibility that the endogenous variables are determined jointly. Usingfour test statistics, Stern and Kaufmann (2000) find that the time series for globalsurface temperature and the radiative forcing of greenhouse gases, anthropogenicsulfur emissions, and solar irradiance contain a stochastic trend. The simplest ex-ample of a stochastic trend is a random walk, which is a discrete time version ofcontinuous time Brownian motion, and is given by Equation (1):

Yt = λYt−1 + εt (1)

in which the autoregressive coefficient λ = 1, and ε is a normally distributedrandom error term (i.e. the innovations) whose mean may be non-zero.

We test the assumption that the time series for emissions and other variablescontain a stochastic trend with the augmented Dickey-Fuller (ADF) test (Dickeyand Fuller, 1979). To carry out the ADF test, we estimate the following regressionfor each variable of interest y:

�yt = α + βt + γ yt−1 +s∑

i=1

δi�yt−i + εt (2)

where � is the first difference operator, t is a linear time trend (which is used torepresent a possible deterministic trend), ε is a random error term, and the coefficientγ = λ − 1.

The null hypothesis of the ADF test is that the series contains a stochastic trend.The ADF test evaluates this null, γ = 0 i.e. λ = 1, by comparing the t-statisticfor γ against a non-standard distribution (e.g. MacKinnon, 1994). Rejecting thenull hypothesis indicates that the autoregressive coefficient (λ) is less than one and

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 251

the series is stationary. If this result is obtained for the level of the series, then theseries is termed integrated of order I(0). If the ADF statistic does not reject the nullhypothesis for the level of the series but rejects the hypothesis for its first difference,the series is said to be integrated of order one I(1) (i.e. it needs to be differencedonce to become stationary). Similarly, a series is integrated of order two I(2) if onlythe second difference of the series is stationary.

The results of the ADF test for the variables in levels in Box 1 fail to reject thehypothesis that the absolute value of the autoregressive coefficient (λ) is equal toone (Table I).

Thus, these variables cannot be analyzed as red noise. Instead, the results indicatethat the time series for temperature, anthropogenic emissions of CO2 and CH4, and

Box 1Model Components

Exogenous Variables Endogenous Variables Identities

ECO2 Anthropogenic carbon emissions(Houghton and Hackler, 1999;Marland and Rotty, 1984)

GLOBL Global SurfaceTemperature (Nichollset al., 1994; Parkeret al., 1998)

RF (prefix) RadiativeForcing CO2, CH9,CFC11, CFC12, N2O(Shine et al., 1991;Kattenberg et al., 1996)

ECH4 Anthropogenic methaneemissions (Kaufmann and Stern,1996)

CO2 Atmosphericconcentrations (Keelingand Whorf, 1994;Etheridge et al., 1996;)

SOX (Wigley and Raper,1992)

CFC Atmospheric concentration ofCFC’s (Prather et al., 1987; Elkinset al., 1994)

CH4 Atmosphericconcentrations(Etheridge et al., 1994;Khalil and Rasmussen,1994; Dlugokenchyet al., 1994)

N2O Atmospheric concentration ofN2O (Prinn et al., 1990, 1995;Machida et al., 1995)

SOX Anthropogenic sulfur emissions(ASL, 1997)

SUN Solar irradiance (Lean et al., 1995)

SOI Southern Oscillation Index (Allenet al., 1991)

NAO Northern Atlantic OscillationIndex (Hurrel, 1995)

RFSS Radiative forcing of stratosphericsulfates in the northern hemisphere(appendix N) or southern hemisphere(appendix S) and latitude (appendix)(Sato et al., 1993)

252 R. K. KAUFMANN ET AL.

their atmospheric concentrations contain a stochastic trend. Traditionally, analysesof the temperature record avoid the assumption of stochastic trends because thesetrends are characterized by their long-term memory – the effects of innovationsdo not fade over time. As such, temperature would be inherently unstable with notendency to return to a long-run mean.

This seeming contradiction is reconciled by identifying the sources of thestochastic trends. The stochastic trends in temperature are caused by stochastictrends in the radiative forcings that drive temperature, and not temperature itself.That is, a direct shock to temperature does not accumulate over time. Rather, thestochastic trends in temperature reflect the stochastic trends in the radiative forcing

Table IUnit Root Tests

Entries are ADF statistics. Sample: 1860–1994

Series s = 2 s = 3 s = 4

I Univariate testsRFAGG −1.82 −1.53 −1.30

�RFAGG −6.32 −5.82 −6.32

RFSSS30N30 −4.43 −4.09 −3.61

�RFSSS30N30 9.36 −8.49 −6.98

CO2 2.36a 0.70 0.54

�CO2 −1.39a −1.23 −1.19

�2CO2 −9.57 −7.49 −6.81

ECO2 0.13a −0.36 −0.20

�ECO2 −3.78 −3.55 −3.26

�ECO2b −4.83 −4.73 −4.51

CH4 −0.89 −0.88 −0.65

�CH4 −6.40 −5.94 −4.78

ECH4 1.06a 0.12 −0.14

�ECH4 −2.68 −2.24 −2.09

SOI −8.25 −6.95 −6.09

NAO −5.89 −5.30 −4.51

II Cointegration tests with estimated coefficients(GLOBL, RFAGG) −5.54 −4.41 −3.98

(CO2, ECO2, GLOBL) −0.12 −0.17 −0.53

(�CO2, ECO2, GLOBL) −4.41 −4.14 −4.21(CO2-.000469ECO2, GLOBL) 0.95 1.71 1.48

(�CO2-.000469ECO2, GLOBL) −3.38 −3.04 −2.98

(CH4, ECH4, GLOBL) −3.98 −3.64 −3.53

(CH4-.3517ECH4, GLOBL) −1.78 −1.26 −1.38

(Continued on next page)

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 253

Table I(Continued)

Series s = 2 s = 3 s = 4

III Cointegration tests with imposed coefficientsCO2-.000469ECO2 2.22 0.76 0.52

�CO2-.000469�ECO2 −1.47 −1.27 −1.19

�CO2-.000469ECO2 −2.81 −2.54 −2.45

�2CO2-.000469�ECO2 −9.49 −7.58 −6.75

CH4-0.3517ECH4 −0.90 −0.81 −0.58

�CH4-0.3517�ECH4 −6.73 −6.30 −5.02

Entries are the ADF test statistics computed using equation (2). Bolded values are significant at the5% level (significance level is computed using MacKinnon’s (1994) approximation for parts I and IIIand using Phillips and Ouliaris (1990) critical values for part II). The linear trend term in equation (2)is excluded, if the series is in first difference (except the case denoted by b). Column heading “s = 2”,etc., indicates the number of lags used in equation (2).aA lag length s is rejected at the 5% level against the alternative s + 1.

of greenhouse gases and anthropogenic sulfur emissions. These trends are like “fin-gerprints” that can be used to identify the effect of radiative forcing on temperature.

Stochastic trends in the radiative forcing data are associated with processes thatare driven by human activity and processes by which the atmosphere accumulatesgases. Anthropogenic emissions of radiatively active gases are determined by eco-nomic activity. The economics literature is replete with studies that indicate GDPand its components contain a stochastic trend therefore, these trends are embod-ied in emissions. The presence of stochastic trends implies that emissions do notincrease as a deterministic function of time (e.g. anthropogenic sulfur emissionsdecrease sharply after the 1970’s due to policies aimed at easing acid depositionand carbon emissions by the Former Soviet Union decline sharply in the 1990’sdue to economic collapse). Therefore, specifying emissions with a deterministictrend is incorrect. Similarly, the long residence time of many radiatively activegases (e.g. CO2, CFC’s) implies that the atmosphere integrates emissions. This canintroduce a stochastic trend in the concentration time series, and the correspondingvalues for radiative forcing. Even if temperature is an inherently stationary series,an autoregressive model that does not control for radiative forcing variables mayhave unit roots.

The presence of stochastic trends invalidates the blind application of standardstatistical techniques such as ordinary least squares (OLS) because they may gener-ate spurious regression results. When evaluated against standard distributions, thecorrelation coefficients and t-statistics for a spurious regression are likely to showthat there is a significant relationship between variables when none exists (Grangerand Newbold, 1974). The potential for spurious regression results led the IPCC tocaution “rigorous statistical tools do not exist to show whether relationships between

254 R. K. KAUFMANN ET AL.

statistically non-stationary data of this kind are truly statistically significant. . ..(Folland et al., 1992, p. 163).”

At about the same time, time series econometricians developed techniques toanalyze relations among integrated time series and thereby avoid spurious regres-sions. These techniques are based on the principle that if two or more integratedtime series have a functionally dependent relation, the stochastic trends presentin some of the series also will be present in the others. This shared trend impliesthat there will be at least one linear combination of the series that is stationaryso that there is no stochastic trend in the residual (i.e. the residual is I(0)). Thisphenomenon is known as cointegration (Engle and Granger, 1987; for a textbooktreatment, see Hamilton, 1994). Kaufmann and Stock (2003) illustrate the notionof spurious regressions and cointegration for carbon cycle data.

Emphasis on cointegration allows us to alleviate some of the difficulties thataccompany the statistical analysis of nonstationary time series which contain con-siderable uncertainty. The uncertainty associated with many of the series in Box 1has been examined explicitly. Marland and Rotty (1984 find that fossil fuel emis-sions have an error of about 10 percent after 1950. Before 1950, the error is about20 percent (Keeling, 1973). Similarly, the temperature data for years after 1900 aremore reliable than data for years prior to 1900 (Jones, 1994). Uncertainty in the icecore data for CO2 concentrations is relatively small 1–3 ppmv (Friedli et al., 1986;Etheridge et al., 1996) but this error is complicated by uncertainty about the date(Craig et al., 1997).

Nonetheless, uncertainty in the time series for CO2 and other variables probablydoes not have a significant effect on the results reported below. If the uncertaintyin the time series is stationary (e.g. white noise), it will not affect tests for cointe-gration and therefore conclusions about the presence of a statistically meaningfulrelation among variables. Alternatively, if the data contain systematic errors thatare stochastically trending, then we will not find any cointegrating relations in thedata because there is no way to eliminate the unobserved stochastic trends asso-ciated with the errors. Under these circumstances, stochastically trending errorswould obfuscate statistical estimates of physically meaningful relationships ratherthan create relationships where none exist. Systematic errors will falsely indicatecointegration only if the same systematic error is present in the time series fortemperature, concentrations, and emissions. Given the very different methods usedto measure and compile the time series for temperature, concentrations, and emis-sions, it is highly unlikely that these time series contain the same trending errors.Thus, it is unlikely that the relations described below are created by stationary ornon-stationary errors in the data.

We estimate the cointegrating relation among integrated variables using Dy-namic Ordinary Least Squares, DOLS (Stock and Watson, 1993). DOLS generatesasymptotically efficient estimates of the regression coefficients for variables thatcointegrate. We use DOLS because it is computationally simple and it performswell relative to other asymptotically efficient estimators (Stock and Watson, 1993).

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 255

The coefficients estimated by DOLS represent the long run relationship amongvariables. DOLS does not estimate the short-run dynamics as this is not necessaryfor asymptotically efficient estimation of the cointegrating relation. The short-runresponses to deviations from the equilibrium relationship are estimated in a secondstage (as described below).

2.1. TEMPERATURE EQUATION

The temperature equation has three goals: (1) to estimate the relationship betweensurface temperature and radiative forcing and the implied temperature sensitivity;(2) to estimate the short-run dynamics by which temperature adjusts to changes inradiative forcing; and (3) to separate the temperature effects of human activity fromthe temperature effects of natural variability. To achieve these goals, the equation forglobal surface temperature is estimated in two stages; (1) the cointegrating relationbetween temperature and radiative forcing and (2) the short run dynamics by whichtemperature adjusts to changes in radiative forcing and patterns of atmospheric andoceanic circulation. To estimate the cointegrating relation between global surfacetemperature and radiative forcing, we compile an aggregate for radiative forcingthat includes the radiative forcing of greenhouse gases (RFCO2, RFCH4, RFCFC11,RFCFC12, and RFN2O), the direct and indirect radiative forcing of anthropogenicsulfur emissions (RFSOX), and the radiative forcing of solar irradiance (RFSUN).This aggregate (RFAGG) includes all components of radiative forcing that containa stochastic trend (Stern and Kaufmann, 2000). To determine whether there is astatistically meaningful relation between this aggregate and global surface tem-perature, we test whether these variables cointegrate. Cointegration is determinedusing OLS to estimate the following equation:

GLOBLt = α + β1RFAGGt + μt (3)

and testing the residual (μt ) for a stochastic trend with the ADF statistic (Equation(2)). If the variables cointegrate, the residual will be stationary. The ADF statisticstrongly rejects (P < 0.01) the null hypothesis that the residual contains a stochastictrend, regardles of the lag length used in Equation (2) (Table I), which indicates thatthe variables in (3) cointegrate. This result is consistent with cointegration betweentemperature and radiative forcing found using different techniques (Kaufmann andStern, 2002). Cointegration indicates that there is a statistically meaningful relationbetween global surface temperature and radiative forcing that can be estimatedefficiently using DOLS. Cointegration also means that this estimate will be efficienteven if global surface temperature and radiative forcing are determined jointly.

The DOLS estimate represents the long-run relationship between temperatureand radiative forcing. As such, temperature does not adjust immediately to changesin radiative forcing. To simulate the rate at which temperature adjusts to changes inradiative forcing and patterns of atmospheric and oceanic circulation, we estimate

256 R. K. KAUFMANN ET AL.

an error correction model (Engle and Granger, 1987):

�GLOBLt = α + β2μt−1 +s∑

i=1

δi�GLOBLt−i +s∑

i=1

φi�RFAGGt−i

+s∑

i=0

πi SOIt−i +s∑

i=0

ψi NAOt−i +s∑

i=0

ζi RFSSt−i + εt (4)

in which μt is the residual from the cointegrating relation estimated by DOLS, SOIis the southern oscillation index, NAO is an index for the North Atlantic Oscillation,and RFSS is the radiative forcing of stratospheric sulfates.

The error correction term (θ2) gives the rate at which temperature adjusts to thetemperature associated with radiative forcing (β2), short run adjustments to tem-perature (δ) radiative forcing (φ), oceanic and atmospheric circulation (π , ψ), andvolcanic activity (ζ ). Equation (4) is estimated using OLS because both the resid-ual from the cointegrating relation and the dependent and independent variablesare stationary. The two step estimation procedure is justified because the DOLSestimate for the cointegrating coefficient converges to its true value faster than OLS(Stock, 1987; Engle and Granger, 1987).

2.2. CONCENTRATION EQUATIONS

Equations for the atmospheric concentration of CO2 and CH4 are derived from anidentity that is based on mass balance in the atmosphere;

xt = ρxt−1 + et + nt (5)

in which xt is the atmospheric concentration at time t and the previous period xt−1,ρ is the retention rate, et denotes net emissions from human sources, and nt denotesnet flows from natural (non-human) sources. Ideally, the autoregressive componentand emissions from natural sources would be modeled using structural equations.Unfortunately, the mechanisms that control atmospheric retention rates and the netrate of natural emissions are uncertain. For example, scientists cannot balance theflow of carbon to and from the atmosphere due to an unknown sink for carbon(for a short review, see Schimel et al., 2001). Because of the unknown carbon sinkand other sources of uncertainty, we approximate natural flows to and from theatmosphere using relatively simple specifications and add complexity (e.g. nonlin-earities, structural changes) to evaluate the degree to which the results are robust.

Although Equation (5) specifies concentrations in absolute levels (e.g. ppm),this mass balance assumes that the pre-industrial atmosphere was in equilibrium.At equilibrium, net natural flows to the atmosphere (nt ) equal the losses associatedwith the retention rate (1−ρ). As the system moves away from equilibrium, changesin temperature and concentrations affect flows to and from the atmosphere. Theseeffects can be differentiated by specifying the retention rate separately from net

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 257

natural flows. The retention rate would be affected most directly by the increase inconcentrations relative to the equilibrium. As atmopsheric concentration increases,the net flow of carbon from the atmosphere to the ocean (and perhaps to the terrestrialbiota) will increase.

The net effect of temperature on natural flows of carbon to the atmosphere inyear t can be approximated (linearly) as follows,

nt = θ tempt + vt (6)

where vt represents natural emissions unrelated to temperature. Due to the lack ofa structural model, the regression coefficient θ represents the net effect of severalphysical mechanisms, some of which may be responsible for the unknown carbonsink. For example, the value of θ from Equation (6) as used for the CO2 equationrepresents the net effect of temperature on net primary production, heterotrophicrespiration, and/or the solubility of carbon dioxide in sea water.

Representing the net effect of temperature on natural emissions suggests a rel-atively simple model for atmospheric concentrations by substituting (6) into (5):

xt = ρxt−1 + et + θ tempt + vt (7)

This specification can be simplified for the CO2 equation by collecting the x termsbased on the assumption that ρ is one. If ρ is one, emissions, concentrations, andtemperature are I(1), and the error is I(0), then (7) is a cointegrating relation. Underthese conditions, the concentration equation for CO2 could be estimated usingDOLS and cointegration would alleviate concerns about simultaneous equationbias.

Unfortunately, this approach is not possible. As indicated in Table I, emissions,concentrations, and temperature do not cointegrate for either methane or carbondioxide. The lack of cointegration is not surprising. If unmodeled natural emissionsare highly persistent, the error term may be I(1). In the case of CO2, the time seriesfor carbon uptake by the unknown carbon sink that is assembled by Houghtonet al. (1998) is I(1) (Kaufmann and Stock, 2003), which implies that vt is I(1).The importance of the unknown carbon sink is indicated by results in Table I,which indicate that CO2, ECO2, and GLOBL may cointegrate, but this possibilitydisappears when we impose mass balance by restricting the coefficient associatedwith ECO2 to 0.000469, which is the physical constant that translates emissionsinto concentrations.

To avoid statistical problems associated with the lack of cointegration, we takethe first difference of Equation (7)

�xt = ρ�xt−1 + �εt + θ�tempt + �vt (8)

Specifying the concentration equation in first differences eliminates all stochastictrends and therefore allows us to avoid the effects of carbon uptake by the unknowncarbon sink(s) and measurement error on statistical estimates for the effect of tem-perature on concentrations.

258 R. K. KAUFMANN ET AL.

Equation (8) can be modified to specify the possibility that the atmosphericretention rate ρ depends on state variables. Specifically, saturation effects couldlead ρ to depend on the ambient concentration of the gas. For example, higher CO2

concentrations slow oceanic uptake via the Revelle effect, which would increase ρ.Alternatively, high concentrations of atmospheric CO2 could increase net primaryproduction via the “CO2 fertilization effect” and thereby increase carbon uptake byterrestrial vegetation. Finally, the retention rate also could depend on temperature.For example, higher temperatures could lengthen the growing season, enhance netprimary production, and thereby increase CO2 uptake by terrestrial vegetation.

To account for possible nonlinearities, we let ρ depend on temperature andconcentrations. If the nonlinearities in ρ are small, state dependence could beapproximated by

�xt = �[ρ(xt−1, tempt−1)xt−1] + �et + θ�tempt + �vt . (9)

For the CO2 equation, we also model �vt (i.e. the net change in natural emissions)as a function of observables zt , specifically SOI (Bacastow, 1976), which we canrepresent as:

�vt = γ ′zt + ut , (10)

where ut is an I(0) error term.Finally, we consider a first order linearization of ρ:

ρ(xt−1, temp′t−1) ∼= ρ0 + ρ1xt−1 + ρ2tempt−1 (11)

Substituting (10) and (11) into (9) and collecting terms yields,

�xt = ρ0�xt−1 + ρ�(x2t−1) + ρ2�(tempt−1xt−1) + �et +

θ�tempt + γ ′zt + ut . (12)

Physical mechanisms imply that surface temperature and the atmospheric con-centration of CO2 and CH4 are determined jointly. For example, temperature ap-pears on the left hand side of the temperature equation (Equation (3)) and appearson the right hand side of the concentration equations (Equations (4) and (5)). Thisjoint determination would cause single equation estimates for the first differencespecifications of the concentration equations to suffer from simultaneous equationbias. This bias could cause point estimates for the effect of temperature on theatmospheric concentration of CO2 or CH4 to overstate their true value, regardlessof the sample size.

From a statistical perspective, the cause of simultaneous equation bias iscorrelation between the endogenous variable (temperature) and the error term. Toavoid the resultant bias, we use an instrumental variable for temperature when tem-perature appears on the right hand side of the concentration equations. The use ofinstrumental variables in the concentration equations can be explained as follows.

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 259

Movements in the atmospheric concentration of carbon dioxide and/or methane arecaused by exogenous variables (e.g. anthropogenic emissions), and endogenousvariables such as temperature. To obtain an unbiased estimate for the effect oftemperature, temperature can be viewed as having two components, an endogenouscomponent, which is associated with changes in CO2 (and CH4) concentrations,and an exogenous component. The latter represents changes in temperature dueto exogenous variables, such as the radiative forcing of stratospheric sulfates (i.e.volcanic activity) or lagged values of endogenous variables. These variables canbe used as instrumental variables because they induce exogenous movements intemperature but do not directly affect the atmospheric concentration of CO2 (orfor the methane equation, CH4). For example, we use RFSS as an instrument fortemperature in the CO2 equation because the radiative forcing of volcanic sulfatesaffects temperature but volcanic activity can be considered exogenous because anyeffect on the atmospheric concentration of CO2 is small and occurs much later thanits effect on temperature (Krakauer and Randerson, 2003). As such, volcanicallyinduced temperature changes generate changes in CO2 concentrations that can beisolated by using RFSS as an instrumental variable for temperature. Following thisapproach, the use of instrumental variables “couples” the statistical estimation of thetemperature and concentration equations in a way that accounts for the simultaneityamong the equations for temperature and concentrations of carbon dioxide andmethane.

To use instrumental variables, the effect of temperature on carbon dioxide ormethane concentrations is estimated in two stages. In the first stage, tempera-ture is regressed on the instrumental variables. This regression equation is usedto generate estimated values for temperature (T ). These estimates are used inplace of the observed values for temperature in the second stage regression (i.e.Equation (8)). Both the first and second stage equations can be estimated us-ing ordinary least squares, which is termed the two stage least squares (2SLS)estimator. Alternatively, coefficients can be estimated using the limited informa-tion maximum likelihood (LIML) estimator (For a description of the the LIMLestimator, see Anderson, 2005; for a textbook treatment of instrumental vari-ables regression, see Wooldridge, 2001). In theory, the LIML and 2SLS es-timators are asymptotically equivalent and have the same asymptotic normaldistribution.

For the instrumental variables method to yield valid statistical inference, it isrequired that the instrumental variables must be correlated with the endogenousseries. The strength of this correlation is evaluated with an F-test on the first stageregression. If the first stage F statistic is less than 10, then the instruments are“weak” and may generate misleading results because the usual t-tests on the secondstage regression coefficients are not accurately approximated by a standard normaldistribution (Staiger and Stock, 1997). To avoid misleading results, confidenceintervals also are estimated by inverting the Anderson-Rubin (Anderson and Rubin,1949) (AR) test statistic.

260 R. K. KAUFMANN ET AL.

3. Results

3.1. TEMPERATURE EQUATION

Estimates and summary statistics for five possible specifications of the long-runrelationship between temperature and radiative forcing are reported in Table II. Inall columns except column 1, the equations are estimated using DOLS. The standarderrors, calculated by both the Newey-West and VARHAC (“Vector AutoregressiveHeteroskedasticity and Autocorrelation Consistent”) procedures, indicate that thereis a statistically meaningful relation between global surface temperature and theaggregate radiative forcing variable that includes greenhouse gases, anthropogenicsulfur emissions, and solar irradiance. As described below, this result is not sensitiveto the lags/leads used by DOLS or the degree to which the components of radiativeforcing are aggregated.

The number of lags and leads used by the DOLS estimator is chosen using theBayes Information Criterion, BIC (Schwarz, 1978). This criterion indicates that

Table IILong-run relation between temperature and radiative forcing: Estimation Results for Equation 3

(1) (2) (3) (4) (5) (6) (7)

Estimationmethod OLS DOLS (1) DOLS (2) DOLS (3) DOLS (1) DOLS (1) DOLS (1)

RFAGG .460 .489∗∗ .511∗∗ .539∗∗ .533∗∗ .428∗∗

[.041] [.044] [.044] [.050] [.057]

{.031} {.029} {.027} {.042} {.055}RFCO2 .321

[.233]

RFCH4 1.92∗∗

[.74]

RFSOX .945∗∗

[.236]

RFSUN .563∗∗

[.243]

Sample 1860–1994 1863–1991 1863–1991 1863–1991 1863–1991 1900–1991 1959–1991

Notes : All regressions are run over the indicated sample period, with earlier and later observationsas initial/terminal conditions. The BIC for regressions (2), (3), and (4) is −2.011, −1.947, −1.898,respectively, so (2) is chosen by BIC. The F-statistic (constructed as the Wald statistic using theNewey-West variance covariance matrix) testing the hypothesis that all the coefficients on the I(1)regressors in (5) are equal is 1.73 (p-value = .15). All regressions include an intercept (not reported).DOLS(p) refers to the Stock-Watson (1993) dynamic OLS estimator with p lags and leads of X, whereX are the I(1) regressors.Standard errors: [ ] = Newey-West (4 lags); {} = VARHAC (3 lags); ( ) = OLS.Coefficients are statistically significantly different from zero at the: ∗∗1%, ∗5%, +10% level.

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 261

one lag/lead is optimal (column 2). This choice does not affect the results. Thepoint estimate (and standard error) for the long-run relation between global surfacetemperature and radiative forcing (β1) changes little if two or three leads/lags areused (compare columns (2), (3), and (4)).

The validity of the confidence interval for β1 constructed using the estimatesand standard errors in Table II depends on radiative forcing having a unit root, thatis, λ = 1 in the notation of (1). If λ is large but not exactly one, then the confidenceintervals can have coverage rates less than the desired 95%, e.g. Kauppi (2004).Moreover, the methods used here are all based on asymptotic theory, which mightnot provide a good approximation for our sample size of 135. At the suggestionof a referee, we therefore conducted a Monte Carlo study in which 135 pairs ofartificial data on (GLOBL, RFAGG) were generated according to the estimatedDOLS equation (Table II, Equation (2)), with auxiliary third order autoregressionsfor the system errors, in which the largest autoregressive root of RFAGG was variedthrough its 95% confidence interval of (0.942, 1.035). The cointegrating coefficientand its standard error were then estimated using DOLS (exactly as in Table II)for 10 000 replications of these artificial data. The results indicate a finite-sampledownward bias of the standard errors of approximately 10%. For λ = 0.942, theasymptotic 95% confidence interval has an actual finite-sample coverage rate of88%; for λ = 1.035, the coverage rate falls to 69%. These results suggest caution ininterpreting the standard errors on RFAGG in Table II, so in our subsequent analysiswe use the more conservative Newey-West standard errors for the cointegratingcoefficient β1.

Consistent with physical theory, the aggregate for radiative forcing is basedon the assumption that the temperature effect of a unit of radiative forcing (e.g.W/m2) is equal across forcings. To test this assumption, we estimate a specificationthat disaggregates the components of radiative forcing. Consistent with statisticalexpectations, individual coefficients are estimated imprecisely (column 5). Thehypothesis of coefficient equality is not rejected at the 10% level (F(4, 108) =1.73, using the Newey and West (1987) covariance matrix, which adjusts for serialcorrelation in the error term (under the maintained assumption of cointegration, thishas an asymptotic distribution which is chi-squared with four degrees of freedom,divided by 4). Together, these results suggest that the results reported in column2 provide the most parsimonious representation of the long-run relation betweentemperature and radiative forcing.

The results of the error correction model indicate that temperature adjusts todeviations from the long run equilibrium relation between temperature and ra-diative forcing. The estimate of the regression coefficient β2 associated with thelagged residual from the cointegrating relation indicates that about 58 percent ofthe disequilibrium in the cointegrating relation between temperature and radiativeforcing is eliminated per year. This rate is similar to the estimate by Kaufmannand Stern (2002). On the other hand, there is no evidence that temperature adjuststo lagged first differences of temperature and radiative forcing (Table III). These

262 R. K. KAUFMANN ET AL.

results are unaffected by the presence of changes in atmospheric and oceanic cir-culation and volcanic activity (columns 3–4) or the number of lags and leads usedby the DOLS estimator (columns 5–6).

Regression results for Equation (4) also show the statistically significant ef-fects of the radiative forcing of stratospheric sulfates, the southern oscillation, andthe North Atlantic oscillation. Consistent with previous results, decreases in theSOI increase global surface temperature (ENSO events raise temperature) whilestratospheric sulfates have a negative effect on surface temperature. The coefficientassociated with stratospheric sulfates is considerably smaller than that associatedwith the aggregate for radiative forcing. This may be caused by the difficulty as-sociated with estimating temperature sensitivity from volcanic forcing. Using sim-ple energy balance models, Lindzen and Giannitsis (1998) demonstrate that largechanges in the parameter for temperature sensitivity have a small impact on thesimulated temperature effect of volcanic activity. For example, the peak tempera-ture effect increases only 0.15 ◦C as temperature sensitivity increases from 0.6 to4.0 ◦C (Lindzen and Giannitsis, 1998). We explore this uncertainty by using thetemperature equation to simulate the temperature effect of the 1991 eruption byMt. Pinatubo (see Simulation Analysis).

The negative sign associated with the variables for NAO contradicts the positivetemperature effect described by Hurrell (1996). This difference may be caused bythe NAO index used (Ponta Delgada, Azores minus Iceland versus Lisbon minusIceland), the use of global surface temperature instead of temperature between 20◦Nand 90◦N, and/or the use of annual averages for temperature and the NAO index ver-sus winter values. For example, the statistical significance of the NAO index dropswhen we use an NAO index that measures the pressure difference between Lisbonand Iceland (column 8, Table III). Consistent with this sensitivity, we do not attachmuch significance to the coefficients associated with the North Atlantic Oscillation.

3.2. CARBON DIOXIDE EQUATION

The small F statistics for the first stage regressions indicate that the instruments fortemperature are weak. The effect of this weakness is evaluated by using 2SLS andLIML techniques to estimate a linear version of Equation (11). A comparison ofcolumns (1) and (2) of Table IV indicates that the 2SLS and LIML point estimatesand standard errors are quite similar. Subsequent specifications are estimated usingLIML because LIML point estimates and confidence intervals are more reliablethan their 2SLS counterparts when the instruments are weak (Staiger and Stock,1997).

Regression results from nearly all specifications indicate that the change in theatmospheric concentration of CO2 is related to global surface temperature (TableIV). As such, these results provide direct evidence that temperature increases since1870 have on net, increased the atmospheric concentration of CO2. The positive

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 263

Tabl

eII

ITe

mpe

ratu

reD

ynam

ics:

Est

imat

ion

Res

ults

for

Equ

atio

n4.

Dep

ende

ntva

riab

le:�

Glo

bl.S

ampl

epe

riod

:186

0–19

95

(1)

(2)

(3)

(4)

(5)

(6)

(7)

DO

LS(

1)D

OL

S(1)

DO

LS(

1)D

OL

S(1)

DO

LS(

2)D

OL

S(3)

DO

LS(

1)

μt−

1−.

598∗∗

(.08

8)−.

614∗∗

(.10

2)−.

582∗∗

(.09

2)−.

553∗∗

(.11

1)−.

593∗∗

(.06

9)−.

623∗∗

(.06

7)−.

526∗∗

(.07

7)

�G

LO

BL

t−1

.057

(.07

7).0

75(.

091)

.048

(.08

6).0

07(.

094)

.029

(.07

3).0

38(.

069)

−.01

7(.

078)

�G

LO

BL

t−2

−.00

2(.

082)

−.06

7(.

074)

�R

FAG

Gt−

1−.

146

(.19

6).0

80(.

226)

−.16

6(.

194)

.016

(.23

5)−.

006

(.21

0)−.

009

(.19

6).0

29(.

229)

�R

FAG

Gt−

2−.

499+

(266

)−.

313

(.24

4)

SOI t

−.05

7∗∗(.

012)

−.05

5∗∗(.

012)

.060

∗∗(.

012)

−.06

4∗∗(.

011)

−.06

2∗∗(.

011)

SOI t

−1−0

.020

(.01

3)−.

022+

(.01

3)−.

019

(.01

4)−.

019

(.01

4)−.

020

(.01

4)

NA

Ot

−.00

5(.

005)

−.00

4(.

005)

−.00

5(.

004)

−.00

4(.

004)

2.2e

-5(3

.53e

-5)

NA

Ot−

1−.

003

(.00

4)−.

002

(.00

4).0

02(.

004)

−.00

2(.

004)

5.03

e-5

(3.8

8e-5

NA

Ot−

2−.

008∗

(.00

4)−.

009∗

(.00

4)−.

009∗

(.00

4)−.

009∗

(.00

4)7.

71e-

5+(4

.36e

-5)

RFS

S t.0

34∗

(.01

4).0

35∗

(.01

4).0

32∗

(.01

4).0

30∗

(.01

4).0

35∗

(.01

5)

Not

es:

The

term

μt

isth

eer

ror

corr

ectio

nte

rmμ

t=

GL

OB

Lt

–β

1R

FAG

Gt,

whe

reth

eco

inte

grat

ing

coef

ficie

ntβ

1w

ases

timat

edby

DO

LS

with

ple

ads

and

lags

of�

RFA

GG

used

inth

ees

timat

ion

equa

tion

(see

the

note

toTa

ble

II).

Het

eros

keda

stic

ity-r

obus

tsta

ndar

der

rors

are

give

nin

pare

nthe

ses.

Coe

ffici

ents

are

stat

istic

ally

sign

ifica

ntly

diff

eren

tfro

mze

roat

the:

∗∗1%

,∗ 5%

,+ 10%

leve

l.

264 R. K. KAUFMANN ET AL.Ta

ble

IVC

arbo

nD

ioxi

deE

quat

ion:

Est

imat

ion

Res

ults

.Dep

ende

ntV

aria

ble:

�C

O2 t

Sam

ple

Peri

od:1

869–

1994

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Met

hod

2SL

SL

IML

LIM

LL

IML

LIM

LL

IML

LIM

LL

IML

�G

LO

BL

t1.

25∗∗

1.36

∗∗1.

38∗∗

1.46

∗∗1.

73∗∗

1.70

∗∗1.

42∗∗

1.46

(.48

)(.

49)

(.50

)(.

54)

(.53

)(.

50)

(.50

)(1

.00)

�E

CO

2 t(×

10−3

).9

71∗∗

(.29

3).9

69∗∗

(.29

4).4

69#

.469

#.4

69#

.469

#.4

69#

.469

#SO

I t.0

80+

(.04

9).0

80+

(.04

9).0

91+

(.05

0).1

10∗

(.05

2).1

08∗

(.05

3).0

97∗

(.04

9).0

95+

(.05

0).0

78(.

075)

�C

O2 t

−1.8

00∗∗

(.05

3).8

01∗∗

(.06

3).8

32∗∗

(.05

1).9

05∗∗

(.03

5)1.

00#

−1.7

0(.

786)

.828

∗∗(.

053)

Stat

eva

riab

lein

tera

ctio

ns:

�(C

O22 t−

1)

.005

(.00

12)

�(G

LO

BL

t−1×

CO

2).0

014

(.00

08)

Tim

edu

mm

yin

tera

ctio

ns:

�C

O2 t

−1<

1910

.781

∗∗(.

159)

�C

O2 t

−1,1

911–

1952

.611

∗∗(.

151)

�C

O2 t

−1,>

1953

.827

∗∗(.

051)

Inte

rcep

t.0

63(.

038)

.063

(.03

9).0

75(.

038)

0#−.

019

(.02

8).6

29(.

436)

.101

∗(.

046)

.078

(.04

0)In

tera

ctio

nF

-tes

tp-v

alue

.13

.25

EC

O2

rest

rict

ion

t-te

st1.

711.

701.

702.

000.

422.

021.

491.

74Fi

rstS

tage

F6.

156.

156.

275.

566.

526.

596.

294.

19J-

test

p-va

lue

.87

.87

.79

.61

.69

.65

.75

.27

AR

Con

fiden

ceIn

terv

al(−

.42,

3.72

)(−

.29,

3.75

)(−

.14,

3.94

)(.

01,4

.06)

(−.2

5,3.

46)

(−.2

5,3.

72)

(−2.

17,5

.99)

Inst

rum

ents

AA

AA

AA

AB

Stan

dard

erro

rsar

egi

ven

inpa

rent

hese

s.T

heA

Rco

nfide

nce

inte

rval

isth

eA

nder

son-

Rub

inco

nfide

nce

inte

rval

;the

Jte

stis

the

test

ofov

erid

entifi

catio

nba

sed

onth

e2S

LS

orL

IML

poin

test

imat

es,d

epen

ding

onth

ees

timat

ion

met

hod

used

fort

heco

lum

n.T

hein

tera

ctio

nF-

test

isth

eF-

test

ofth

ehy

poth

esis

that

the

coef

ficie

nts

onth

ein

tera

ctio

nva

riab

les

are

both

zero

.The

EC

O2

rest

rict

ion

t-te

stis

the

t-te

stof

the

hypo

thes

isth

atth

eco

effic

ient

on�

EC

O2

equa

ls.0

0046

9;fo

rthe

rest

rict

edre

gres

sion

s,th

isis

the

LIM

Lt-

stat

istic

from

the

corr

espo

ndin

gun

rest

rict

edre

gres

sion

with

the

rest

ofth

esp

ecifi

catio

not

herw

ise

the

sam

e.T

hein

stru

men

tare

:In

stru

men

tset

A:�

RFS

SN30

S30,

�R

FSSS

3090

,SO

I t−2

,SO

I t−3

,�G

LO

BL

t−2,�

GL

OB

Lt−

3In

stru

men

tset

B:�

RFS

SN30

S30,

�R

FSSS

3090

#co

nstr

aine

dva

lue.

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 265

effect may represent several physical mechanisms, such as a reduction in the ocean’sability to absorb carbon (Macintyre, 1978), changes in upwelling that slow the flowof carbon from the ocean to the atmosphere (Dettinger and Ghil, 1998), and/or anincrease in heterotrophic respiration relative to net primary production (Vukicevicet al., 2001). The regression coefficients indicate that a 1 ◦C rise in temperatureincreases the atmospheric concentration of CO2 by about 1.5 ppmv. Although thiseffect does not reflect any single physical mechanism, the magnitude of this net ef-fect is similar with a theoretical analysis of the ocean’s ability to absorb CO2, whichindicates that a 1 ◦C rise in ocean surface temperature (100m surface layer) would in-crease the atmospheric concentration of CO2 by 1.5 ppm (Macintyre, 1978). The re-gression coefficient also is the same order of magnitude of an empirical estimate thatsuggests CO2 concentrations rise by about 3 ppmv per 1 ◦C (Keeling et al., 1989).

Despite the consistency of the temperature effect, the regression coefficientsassociated with concentrations and emissions reflect the inability to balance carbonflows in and out of the atmosphere due to the unknown carbon sink. Mass balancein Equation (5) implies that there is a unit coefficient on human emissions and nointercept. A unit coefficient on et in Equation (5) corresponds to the restriction thatthe coefficient associated with �ECO2 is 0.000469, which is the conversion factorthat translates a mass of CO2 (thousand petagrams) into its atmospheric concen-tration (ppmv). Although the estimated coefficient is twice this, this estimate isimprecise and the hypothesis that the coefficient on �ECO2 equals its theoreticalvalue is not rejected at the 5% level in specifications (1)–(3). To compensate for thislarge value, the statistical technique assigns a low value to ρ. This combination ofvalues “solves” the difficulties associated with the unknown carbon sink by puttingextra of carbon into the atmosphere and then taking it out quickly. In column 3, weensure mass balance by imposing a value of 0.000469 on the coefficient associatedwith �ECO2. The value of ρ rises slightly and the intercept becomes statisticallysignificant (p < 0.05). Taken literally, this intercept increases the atmospheric con-centration of CO2 by 0.075 ppmv annually. This secular increase is nonsensical andso we suppress the constant and impose mass balance on emissions in specification(5). Together, these restrictions are inconsistent with the inability to balance theglobal carbon cycle due to the unknown carbon sink, therefore the mass balancerestriction on the coefficient associated with �ECO2 is rejected (p < 0.05).

Statistical estimates for ρ are substantially lower (i.e. shorter residence time)than implied by the physical science evidence. This result reflects inadequacies inexisting statistical techniques. Estimators of autoregressive coefficients are biasedtowards zero (Hamilton, 1994), and this bias is especially pronounced when thetrue autoregressive coefficient is large (i.e. when ρ is close to 1). This bias increaseswhen a constant is included, and this makes the estimated value for ρ smaller inspecification (3) than in (4). This bias can be eliminated in simple models but to thebest of our knowledge, methods that can adjust for this bias have not been developedfor the type of models estimated here, in which there are additional regressors andthe estimation procedure uses instrumental variables methods.

266 R. K. KAUFMANN ET AL.

To evaluate the effect of this bias on the statistical estimates for the temperatureeffect, we impose different values on ρ. As an extreme case, in which there is no netflow of carbon from the atmosphere, we impose ρ = 1. This extreme assumptiondoes not materially alter the estimated coefficients on the other variables. Nor can wereject a specification that imposes a value of 0.965 for ρ, which is the value obtainedby OLS estimation of Equation (7) in levels when we impose the mass balanceconversion coefficient 0.000469 on ECO2. Such a value for ρ is consistent with thelong persistence of a one-time pulse of carbon indicated by models that simulatethe physical mechanisms that determine the rate of carbon flows to and from theatmosphere (Albritton and Filho, 2001). Based on these results, we conclude that therelatively low point estimate of ρ in specifications (1)–(3) is caused by non-normaland biased estimator distributions. As a result, we cannot use the estimates of ρ tomake statements about the size of the CO2 fertilization effect, the Revelle effect, orany other physical mechanism by which the atmospheric concentration of CO2 mayaffect the rate of which carbon flows to or from the atmosphere. Nonetheless, thesedifficulties do not affect the estimates of the other coefficients, most importantlythe effect of temperature on the atmospheric concentration of CO2.

Despite uncertainty about the value of ρ, there is little statistical evidence that ρ isstate dependent. The nonlinear terms in column 6 are jointly insignificant at the 10%level, although the temperature-concentration term is significant at the 10% (butnot 5% level). Regression results from a specification that represents the possibilitythat the removal rate changes over time show no marked trend (specification 7),and the null hypothesis that ρ is constant cannot be rejected (p > 0.25).

Specification (8) considers a reduced instrument list, which includes the radia-tive forcing of stratospheric sulfates only. We drop lags of �GLOBL and SOIbecause they may be correlated with a serially correlated error term. Withoutlags of �GLOBL and SOI, the first stage F statistic drops to 4.19, and the es-timates are considerably less precise. Nonetheless, the point estimates are similarto those in column (3), which suggests that the conclusion that temperature affectsthe flow of carbon to and from the atmosphere is not sensitive to the instrumentset.

3.3. METHANE EQUATION

The mass balance equation seems inconsistent with the time series properties ofmethane concentrations, which appear to be I(1), and anthropogenic emissions,which appear to be I(2). The seeming contradiction can be reconciled by themagnitude of anthropogenic emissions and the physical determinants of atmo-spheric methane concentrations. In any year, anthropogenic methane emissions area relatively small component of atmospheric concentrations, which implies thatthe I(2) component of atmospheric concentrations is small. Furthermore, the at-mospheric lifetime of methane is relatively short (the autoregressive coefficient is

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 267

significantly less than 1.0) which would tend to “erase” the I(2) trend in methaneemissions. Together, these two factors make it difficult to detect the I(2) trend in theconcentration data, therefore CH4 appears to be I(1). Because of this ambiguity, weexamine the time series properties of CH4 − 0.3517 × ECH4 (0.3517 is the conver-sion factor that translates teragrams of CH4 to ppbv). As indicated in Table I, thisconstructed series is I(1). This is consistent with a retention rate (ρ) substantiallyless than one, GLOBL being I(1), and an error term that is either I(0) or I(1). Avalue of ρ substantially less than one is consistent with the short residence time ofmethane (about a decade) relative to carbon dioxide.

The first stage F for the CH4 equation is somewhat larger than for the CO2 equa-tion, and the LIML and 2SLS results are quite similar (Table V). None of the specifi-cations are rejected by the overidentification J-test, and all of the AR confidence in-tervals for the coefficient on �GLOBL are nonempty. The AR confidence intervalsgenerally are similar to, but somewhat wider than, the LIML confidence intervals.

For all specifications, the point estimates for the effect of global surface temper-ature on methane concentrations are similar (Table V). Despite this similarity, theseestimates are imprecise. The hypothesis that the coefficient on �GLOBL is zerocannot be rejected at the 10% level using the LIML estimates, and the AR confi-dence intervals for this coefficient all include zero. Together, these results indicatethat we cannot estimate the effect of global surface temperature on the atmosphericconcentration of methane in a statistically precise manner.

Estimation results for specifications (1)–(3) are consistent with mass balance.Although the point estimate of the coefficient on �ECH4 is almost three times itstheoretical value 0.3517, this estimate is very imprecise and the hypothesis that itequals its theoretical value is not rejected at the 10% level. Similarly, the interceptsin specifications (1) and (2) are not statistically different from zero at the 10%level. Finally, the estimates for ρ vary between 0.35 and 0.44. These values aresignificantly less than 1.0, which is consistent with physical theory. On the otherhand, the values imply a residence time that is shorter than the 8.4 year atmosphericresidence time and the 12 year perturbation time reported in the literature (Ehhaltand Prather, 2001).

The physical mechanisms that remove methane from the atmosphere imply thatρ is not constant, but results that indicate ρ is not constant probably are statisticallyspurious artifacts because they contradict physical theory. The only statisticallysignificant nonlinear effect in specification (4) is the concentration term. The neg-ative sign on this term corresponds to an inverse saturation effect (i.e. a smallervalue of ρ at higher concentrations). Moreover, estimates for the interaction termsin specification (5) suggest that the removal rate increases from 37% (1–0.629) inthe first third of the sample to 78% (1–0.217) in the final third. There is no phys-ical reason to believe that the direction or magnitude of this change is plausible.Instead, this pattern of time varying persistence may be caused by changes in thedata. Concentration data change smoothly early in the sample period when valuesare derived from ice cores, as opposed to sharp changes during the observational

268 R. K. KAUFMANN ET AL.Ta

ble

VC

H4

Equ

atio

n:E

stim

atio

nR

esul

ts.D

epen

dent

Var

iabl

e:�

CH

4 tSa

mpl

ePe

riod

:186

9–19

95

(1)

(2)

(3)

(4)

(5)

Met

hod

2SL

SL

IML

LIM

LL

IML

LIM

L

�G

LO

BL

t21

.621

.024

.424

.920

.2

(17.

8)(1

8.7)

(19.

8)(1

8.1)

(18.

7)

�E

CH

4 t.9

01.9

01.3

517#

.351

7#.3

517#

(.57

4)(.

574)

�C

H4 t

−1.3

51∗∗

(.08

3).3

51∗∗

(.08

3).4

40∗∗

(.07

6)1.

337∗∗

(.48

9)

Stat

eva

riab

lein

tera

ctio

ns:

�(C

H42

t−1)

−.00

042∗

(.00

021)

�(G

LO

BL

t−1×

CH

4 T−1

).0

0692

(.00

986)

Tim

edu

mm

yin

tera

ctio

ns:

.629

�C

H4 t

−1<

1910

(.38

4)

�C

H4 t

−1,1

911–

1952

.450

(.10

4)

�C

H4 t

−1,>

1953

.217

(.12

5)

Inte

rcep

t2.

00(1

.85)

2.00

(1.8

5)0#

−3.2

4(3

.45)

3.58

∗(1

.41)

Inte

ract

ion

F-t

estp

-val

ue.0

4.0

8

EC

H4

rest

rict

ion

t-te

st.9

6.9

6.9

62.

33*

1.05

Firs

tSta

geF

8.23

8.23

7.41

9.26

8.12

J-te

stp-

valu

e.5

3.5

3.2

0.4

7.4

9

AR

Con

fiden

ceIn

terv

al(−

46.6

,86.

1)(−

29.0

,77.

1)(−

36.9

,86.

1)(−

46.4

,84.

1)

Inst

rum

ents

CC

CC

C

See

the

note

sto

Tabl

eIV

.The

inst

rum

ents

are:

Inst

rum

ents

etC

:�R

FSSN

30S3

0,�

RFS

SS30

90,S

OI t

,SO

I t−1

,SO

I t−2

,SO

I t−3

,�G

LO

BL

t−2,�

GL

OB

Lt−

3.

#co

nstr

aine

dva

lue.

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 269

record. Smoothness will increase the estimates for persistence, which will increaseρ early in the sample.

As a final diagnostic, the residuals from the three equations are modeled using asecond order vector autoregression. There is a small but nonzero serial correlationin each equation, which is consistent with unmodeled nonlinearities and omittedserially correlated influences on net natural emissions. However, there is no crosscorrelation (p > 0.15), which suggests that these omitted variables do not masksome remaining simultaneity that is not modeled explicitly.

4. Simulation Analysis

We assemble the four endogenous equations with the identities to generate a simplesimultaneous equation model as follows:

Endogenous equations

GLOBLt = 0.489RFAGGt (13)

�GLOBLt = −0.582(GLOBLt−1 − 0.489 RFAGGt−1) + 0.034 RFSSt

+ 0.047�GLOBLt−1 − 0.166 �RFAGGt−1 − 0.057 SOIt

−0.020SOIt−1 − 0.005 NAOt − 0.003 NAOt−1 − 0.008 NAOt−2

(14)

�CO2t = 0.000469�ECO2t + 1.46�GLOBLt−1 + 0.110 SOIt−1

+0.905�CO2t−1 (15)

�CH4t = 0.3517ECH4t + 24.4�GLOBLt−1 + 0.44�CH4t−1 (16)

Identities

GLOBLt = GLOBLt−1 + �GLOBLt (17)

CO2t = CO2t−1 + �CO2t (18)

CH4t = CH4t−1 + �CH4t (19)

RFCO2t = 6.3 ln(CO2t/CO2 1860) (20)

RFCH4 = 0.03873(√

CH4t −√

CH4 1860

)(21)

RFAGGt = RFCO2t + RFCH4t + RFCFC11t + RFCFC12t + RFN2Ot

+SOXt + RFSUNt (22)

Exogenous variables

RFCFC11, RFCFC12, RFN2O, RFSUN, SOX, ECO2, ECH4

in which parameters are taken from the following estimation results [Equation (13)Table II specification (2); Equation (14), Table III specification (3); Equation (15)

270 R. K. KAUFMANN ET AL.

Table IV specification (4); Equation (16) Table V specification (3). In this section,we use this model to: (1) separate the effects of natural variability and humanactivity on global surface temperature between 1870 and 1990; (2) estimate theeffect of exogenous variables on temperature and concentrations of atmosphericCO2, and (3) evaluate temperature sensitivity and climate dynamics.

4.1. THE TEMPERATURE EFFECT OF NATURAL VARIABILITY

VERSUS HUMAN ACTIVITY

To separate the temperature effect of human activity from natural variability, wesimulate the simple climate model given by Equations (13)–(22) with either humanactivity or natural variability held constant. To assess the effect of natural variability,we hold human activity constant by setting anthropogenic emissions of CO2, CH4,and sulfur to zero, set the concentration of CO2, CH4, CFC11, CFC12, and N2Oto their 1870 level, and simulate the model with the historical pattern of changesin SUN, RFSS, SOI, and NAO. Changes in these variables cause a great deal ofvariability, but do not cause global surface temperature to increase significantly(Figure 1).

To asses the effect of human activity, we eliminate the effects of natural vari-ability. We do so by holding SUN, RFSS, SOI, and NAO at their 1870 level. To

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990

Deg

rees

Cel

sius

Figure 1. Historical temperature (solid line), temperature simulation based on changes in naturalforces (RFSUN, RFSS, SOI, and NAO–dashed-dotted line), temperature simulation based on changesin radiative forcing associated with human activity (RFCO2, RFCH4, RFCFC11, RFCFC12, RFN20,RFSOX – dotted line), temperature simulation based on both natural factors and radiatively activegases associated with human activity (dashed line).

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 271

avoid the disruptive effect of the unknown carbon sink and the relatively short res-idence time for methane on the model’s ability to simulate the temperature effectassociated with anthropogenic emissions of CO2 and CH4, we set their concentra-tions exogenously consistent with their historical values (rather than use Equations(15) and (16)). The concentrations of CFC11, CFC12, and N2O also follow theirhistorical values.

Simulation results indicate that changes in the atmospheric concentration ofgreenhouse gases and anthropogenic sulfur emissions account for much of theincrease in global temperature between 1870 and 1990 (Figure 1). This increaseis not steady. Temperature increases between 1910 and 1944 and from 1970 to1990 are associated with increases in total radiative forcing. These increases areassociated with an increase in the radiative forcing of greenhouse gases relativeto anthropogenic sulfur emissions. The radiative forcing of anthropogenic sulfuremissions increases at about the same rate as greenhouse gases between 1944 and1976. As a result, there is relatively little net increase/decrease in total radiativeforcing and therefore, global surface temperature. The timing of these temperatureeffects is consistent with results obtained from model simulations (Andronovaand Schlesinger, 2000; Tett et al., 1999). Finally, an experiment that simulatesthe historical changes in both natural variables and gases associated with humanactivity is able to account for much of variation in global temperature over the last130 years (Figure 1).

4.2. THE EFFECT OF EXOGENOUS VARIABLES

The effect of exogenous variables on temperature and concentrations can be as-sessed by using the model to simulate impulse response functions. To run theseexperiments, we set exogenous variables to their pre-industrial value (e.g. zero emis-sions of CO2, CH4, and sulfur) or zero (e.g. SOI, NAO, RFSS), use a version of theCO2 equation that suppresses the constant (specification 3), and allow the system tocome to equilibrium. We simulate a series of experiments in which this equilibriumis perturbed by a one-time increase in CO2 or CH4 emissions (9.6 Pg. and 580 Tg.respectively) that generates an immediate 0.1 W m−2 increase in radiative forcing(the CO2 and CH4 experiment respectively), a one-time 0.1 W m−2 increase in theradiative forcing of solar irradiance (sun experiment), a two year decrease in the SOIthat mimics the 1982–1983 El Nino (ENSO experiment), and a three year changein RFSS that mimics the eruption of Mount Pinatubo (Pinatubo experiment).

Consistent with the equal impact of across forcings, the one-time increasein CO2 emissions, CH4 emissions, or solar irradiance has the same effect ontemperature in the short run (Figure 2). The long-run effect of these changes variesby forcing. Consistent with our argument that temperature itself is not I(1), theincrease in solar activity has little effect beyond the first year. The temperatureeffect of the one-time increase in CH4 emissions is slightly longer due to the

272 R. K. KAUFMANN ET AL.

Figure 2. The temperature effects of the CO2 experiment (long dashed line), the CH4 experiment(dotted line), the sun experiment (short dashed line), the ENSO experiment (solid line), and thePinatubo experiment (dashed doted line).

persistence of CH4 in the atmosphere. As described earlier, the persistence ofthis effect probably is too short because the CH4 equation underestimates theatmospheric residence time of methane. Because the atmospheric lifetime ofCO2 is significantly longer than the rate at which temperature adjusts to theincrease in radiative forcing, temperature continues to rise beyond the immediateeffect. Nonetheless, this effect also fades over time. Consistent with the statisticaldifficulties associated with estimating ρ for the CO2 equation, the length of thisperiod cannot be determined with much certainty.

The impulse response functions indicate that the El Nino-Southern Oscillationhas a significant effect on global surface temperature and atmospheric CO2. Valuesfor the SOI which simulate the 1982–1983 El Nino event increase global surfacetemperature by about 0.1 ◦C (Figure 2). This effect is slightly lower than previousestimates of about 0.2 ◦C (Jones, 1989; Angell, 1988). This temperature increaseis not large enough to offset completely the effect of the SOI variable in the CO2

equation therefore, the ENSO experiment indicates that on net, the 1982–1983El Nino reduced atmospheric CO2 by about 0.34 ppmv. This negative effect isconsistent with analyses that suggest that ENSO events reduce the flow of carbonto the atmosphere by slowing ocean upwelling (Dettinger and Ghil, 1998; Winguthet al., 1998) or that enhanced incoming solar radiation during the early stages ofan ENSO event increase carbon uptake by terrestrial vegetation (Graham et al.,2003; Yang and Wang, 2000). This reduction also is consistent with an inverse

EMISSIONS, CONCENTRATIONS, & TEMPERATURE: A TIME SERIES ANALYSIS 273

relation between ENSO and the atmospheric growth rate of CO2 (Rayner et al.,1999; Francey et al., 1995). Conversely, the net negative effect seems to contradictresults that indicate atmospheric concentrations of CO2 rise during ENSO eventsbecause reduced uptake by the terrestrial biota exceeds the increased uptake by theoceans (Joos et al., 1995; Keeling et al., 1995).

Consistent with the uncertainty about the temperature effect of volcanic activitydescribed above, estimates for the temperature effect of the Mount Pinatubo eruptionvary. The Pinatubo experiment indicates that this eruption reduced annual globalsurface temperature by a maximum of −0.12 ◦C (Figure 2). This effect is similar tothe two year reduction (−0.19 and −0.18 ◦C) found by Yang and Schlesinger (2001)after they removed the temperature effects of the concurrent El Nino–SouthernOscillation. Both of these effects are slightly smaller than the temperature effectsgenerated by Hansen et al. (1992). Together, these results imply that our statisticalestimate cannot be used to narrow the existing range for the temperature effect ofstratospheric sulfates. Nonetheless, the statistical estimate for the temperature effectof stratospheric sulfates is not unreasonable and does not undermine the estimatefor the temperature effect of greenhouse gases, anthropogenic sulfur emissions, andsolar irradiance.

4.3. TEMPERATURE SENSITIVITY AND TEMPERATURE DYNAMICS

The estimates for β1 in Equation (3) and β2in Equation (4) can be used to evaluate thetemperature sensitivity and dynamics of the climate system. The DOLS estimate ofβ1 reported in column 2 of Table II (0.489) indicates that doubling the pre-industrialconcentration of CO2 would increase global surface temperature by about 2.1 ◦C(0.489 × 6.3 × ln(2)), with a 95% confidence interval of 1.8 to 2.5 ◦C based on themore conservative Newey-West standard errors in Table II.

This estimate begs the question regarding the time scale of the change. Theseadjustments are summarized by three definitions of temperature sensitivity: thetransient climate response, the equilibrium climate sensitivity, and the effectiveclimate sensitivity (Cubasch and Meehl, 2001). Analysis of simulations run forthe coupled models intercomparison project indicate that the temperature effectof doubled CO2 estimated here is consistent with the transient climate response(Kaufmann et al., 2006).

Consistent with this interpretation, the temperature sensitivity implied by β1

2.1 ◦C falls in the middle of the 1.2 to 3.1 ◦C range of values for the transientclimate response simulated by climate models (Cubasch and Meehl, 2001). Thisrange can be narrowed by our estimate for β1. Using the Newey West estimator,which generates larger standard errors than the VARHAC estimator, the 95 percentconfidence interval for our statistical estimate of β1 is equivalent to a temperaturesensitivity of 1.8–2.5 ◦C. This range does not vary greatly by period. The DOLSestimate forβ1 with data for 1900–1991 (the more reliable portion of the temperature

274 R. K. KAUFMANN ET AL.

data), yields a 95 percent confidence interval of 1.9–2.8 ◦C while data for 1959–1991 (the start date for the Mauna Loa record for atmospheric CO2 measurements)yields a range of 1.4–2.4 ◦C.

The rapid rate of adjustment represent by β2 also is consistent with the rates ofadjustment associated with the transient climate response. Simulating the CMIP2experiments analyzed by Kaufmann et al. (2006) (an annual one percent increase inatmospheric CO2 until concentration doubles) with Equations (13)–(22) indicatesthat about 95 percent of the temperature increase implied by β1 occurs at the timethat the atmospheric concentration of CO2 doubles. This rapid rate of adjustmentand sharp slowdown thereafter is consistent with the abrupt slow-down in the tem-perature increase simulated by the CMIP2 experiments that occurs immediatelyafter the initial doubling of the atmospheric concentration of CO2.

5. Conclusion

Recent advances in time series econometrics can be used to estimate statisticallymeaningful equations for the relation among human activities that emit CO2 andCH4, the atmospheric concentration of these gases, and global surface temperature.The results provide direct evidence that there is a statistically meaningful relation-ship between global surface temperature and an aggregate of radiative forcing thatincludes greenhouse gases, anthropogenic sulfur emissions, and solar activity. Asimple model based on these results indicates that greenhouse gases and anthro-pogenic sulfur emissions are largely responsible for the observed increase in globalsurface temperature between 1870 and 1990. This result is direct evidence for theeffect of human activity and global climate.

The effect of human activity on surface temperature is reinforced by the simulta-neous relationship between surface temperature and the atmospheric concentrationof CO2. Our results indicate that the global carbon cycle contains a positive feed-back loop in which temperature increases associated with human activities thatemit CO2 (and other greenhouse gases) change flows to and from the atmospherein a way that on net increases the atmospheric concentrations of CO2, increasesits radiative forcing, and increases temperature further. Together, our estimates forthe simultaneous linkages among climate, human activity, and the biogeochemicalcycling of carbon improve empirical estimates that focus on individual links andomit important variables.

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(Received 14 June 2004; in revised form 21 November 2005)